# How do you condense Ln 3 − 2 ln(8 + 4) ?

Aug 1, 2018

$- \ln \left(48\right)$

#### Explanation:

We have to remember two different log rules:
Rule 1: $a \log b = \log \left({b}^{a}\right)$
Rule 2: $\log \left(a\right) + \log \left(b\right) = \log \left(a b\right)$
(Note: Rule 1 is actually a specific case of Rule 2 if $a = b$ and you do it a bunch of times)

From that, we can simplify the original:
$\ln \left(3\right) - 2 \ln \left(8 + 4\right)$
$\ln \left(3\right) - 2 \ln \left(12\right)$ (by addition)
$\ln \left(3\right) + \ln \left({12}^{- 2}\right)$ (by rule 1)
$\ln \left(3 \cdot {12}^{- 2}\right)$ (by rule 2)
$\ln \left(\frac{3}{144}\right)$ (by definition)
$\ln \left(\frac{1}{48}\right) = - \ln \left(48\right)$